Question: Solve for $x$ and $y$ using elimination. $\begin{align*}7x-5y &= 3 \\ -9x+3y &= 3\end{align*}$
Explanation: We can eliminate $y$ when its corresponding coefficients are negative inverses. Recalling our knowledge of least common multiples, multiply the top equation by $3$ and the bottom equation by $5$ $\begin{align*}21x-15y &= 9\\ -45x+15y &= 15\end{align*}$ Add the top and bottom equations. $-24x = 24$ Divide both sides by $-24$ and reduce as necessary. $x = -1$ Substitute $-1$ for $x$ in the top equation. $7( -1)-5y = 3$ $-7-5y = 3$ $-5y = 10$ $y = -2$ The solution is $\enspace x = -1, \enspace y = -2$.